{"id":"d185231b-766a-4bd1-85c7-6f18a6725dba","arxiv_id":"2411.17169","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims two nonnegative solutions for a mixed local/nonlocal Kirchhoff equation with critical growth and sign-changing weight, but the proof has notable gaps.","lead":"This paper claims that a mixed local and nonlocal Kirchhoff equation with a sign-changing concave term and critical exponent has two nonnegative solutions for small parameters. The proof uses standard Nehari-manifold machinery, but several load-bearing steps are not supported as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.5 defines the Nehari-constraint map without the Kirchhoff coefficient, so Lemma 3.6's Palais-Smale sequence and both existence proofs are unsupported.","rationale":"The paper aims to prove two nontrivial nonnegative solutions for a mixed local/nonlocal Kirchhoff problem with critical growth and sign-changing weight, using the Nehari manifold and a Brezis-Lieb compactness argument. The proof chain has three pillars: the Nehari decomposition and a Palais-Smale sequence, the first solution via negative energy in N^+_lambda, and the second solution via an energy estimate below the critical threshold. The weakest pillar is Lemma 3.5 because its auxiliary map F_u drops the Kirchhoff terms; the displayed derivative (3.3) is the derivative for the purely local problem with a=1 and b=0, not for the stated functional. Lemma 3.6 is the immediate consequence of Lemma 3.5, and both main theorems invoke Lemma 3.6. Therefore the variational argument lacks a proven foundation as written. The reader's named weakest assumption, f > 0 on the support of the concentrating test function, is a real and independent obstruction to the second-solution proof in Proposition 5.2, but it appears later in the argument and only affects Theorem 1.2. I therefore partially agree with the reader: the overall rejection is correct, but the more load-bearing defect is the algebraic mismatch in Lemma 3.5. If only the f-positivity issue were fixed, Lemma 3.5 would still block both existence proofs; if only Lemma 3.5 were fixed, the unstated f-positivity hypothesis would still block the second solution. The manuscript also has other issues, but this one suffices to invalidate the central claim as written. Since the reader already recommended rejection and my analysis does not alter that conclusion, the verdict remains unchanged.","tokens_in":16097,"tokens_out":6470,"duration_ms":60959,"concrete_test":"Independently recompute Lemma 3.5 with the actual Nehari derivative: set F_u(t,v) = (1/t) <J'_lambda(t(u-v)), t(u-v)> or equivalently the full Kirchhoff expression involving a t^2 rho(w)^2 + b t^{2theta} rho(w)^{2theta}, and evaluate ∂F_u/∂t at (1,0). If the derivative is (2-p)a rho(u)^2 + (2theta-p)b rho(u)^{2theta} - (2*-p)∫|u|^{2*}, not the value shown in (3.3), then Lemma 3.5 as printed is false and Lemma 3.6 lacks proof. This single re-derivation settles whether the Palais-Smale sequence used in both theorems is actually supported by the manuscript.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing defect is in Lemma 3.5, whose auxiliary map F_u does not encode the Kirchhoff problem. For w = u - v, the condition t w in N_lambda is a t^2 rho(w)^2 + b t^{2theta} rho(w)^{2theta} - lambda t^p ∫ f |w|^p - t^{2*} ∫ |w|^{2*} = 0. Lemma 3.5 instead sets F_u(t,v) = t^2 rho(u-v)^2 - lambda t^p ∫ f |u-v|^p - t^{2*} ∫ |u-v|^{2*}, omitting both the a-term and the b-term. Consequently the derivative formula (3.3) has denominator (2-p)rho(u)^2 - (2*-p)∫|u|^{2*}; with the correct F_u the denominator must contain (2-p)a rho(u)^2 + (2theta-p)b rho(u)^{2theta} - (2*-p)∫|u|^{2*}. Thus the implicit-function argument does not produce maps with ξ(v)(u-v) in N_lambda for the stated functional, and Lemma 3.6's conclusion that a minimizing sequence can be chosen with J'_lambda(u_k) = o_k(1) does not follow. Both Theorem 1.1 and Theorem 1.2 use Lemma 3.6 to obtain a Palais-Smale sequence, so without this step neither existence proof is complete. This is an internal inconsistency, not merely a matter of choosing between known alternatives. Separately, the reader's f-positivity point about Proposition 5.2 is also valid: the step 'Using f > 0 in the support of u_{epsilon,eta}' makes inequality (5.5) depend on an assumption not stated in Theorem 1.2, so the second-solution energy estimate is not established as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a mixed local/nonlocal Dirichlet problem (P_lambda) with Kirchhoff coefficient M(t)=a+b t^{theta-1}, a sign-changing concave term lambda f|u|^{p-2}u, and a critical Sobolev term. The main results claim one nontrivial nonnegative solution for small lambda (Theorem 1.1) and, under N+4s<6 and b sufficiently small, two distinct nonnegative solutions (Theorem 1.2). The method is Nehari-manifold minimization with fibre maps, an extremal parameter lambda_*, a compactness threshold c_lambda, and energy estimates based on the sharp mixed Sobolev constant from [11]. The paper is clearly organized, and the proposed compactness threshold is explicit.","tokens_in":16408,"tokens_out":17348,"duration_ms":155458,"significance":"If the proof were complete, the work would be a useful extension of Ambrosetti-Brezis-Cerami type results to mixed local/nonlocal Kirchhoff problems, and the use of the non-achieved mixed Sobolev constant [11] is a relevant technical novelty. The paper states its hypotheses and threshold parameters precisely and gives an explicit lambda-dependent compactness level in Proposition 3.7. However, the proof currently contains several load-bearing gaps: the implicit-function argument in Lemma 3.5 does not encode the Kirchhoff terms, the second-solution estimate in Proposition 5.2 uses an unstated positivity assumption on f, and the nonnegativity conclusion in Section 4 is not derived from a critical point of the functional that was minimized. These issues are localized and appear repairable, but they block acceptance in the present form.","major_comments":[{"comment":"The auxiliary function F_u(t,v) used for the implicit function theorem is t^2 rho(u-v)^2 - lambda t^p int f|u-v|^p - t^{2*} int |u-v|^{2*}, which is the Nehari equation for the semilinear problem with M(t) equivalent to 1, not for (P_lambda). For the problem at hand the Nehari condition for t(u-v) is a t^2 rho(w)^2 + b t^{2theta} rho(w)^{2theta} = lambda t^p int f|w|^p + t^{2*} int |w|^{2*}. Consequently the denominator in (3.3) should contain (2-p)a rho(u)^2 + (2theta-p)b rho(u)^{2theta} - (2*-p) int |u|^{2*}, and the numerator should include the corresponding a- and b-terms. Because Lemma 3.6 is the step that produces a Palais-Smale sequence and is used in both Theorems 1.1 and 1.2, the existence proofs are not supported as written. This is repairable by inserting the correct F_u, but it is not a purely cosmetic typo.","section":"Section 3, Lemma 3.5 and Eq. (3.3)"},{"comment":"The reduction from (5.4) to (5.5) uses 'f > 0 in the support of u_{epsilon,eta}', but this is nowhere assumed in Theorem 1.2 or Proposition 5.2: the standing assumption is only f in L^{2*/(2*-p)} and sign-changing. Since u_{epsilon,eta} is supported in a ball around 0, this is an additional pointwise positivity assumption on f near 0. The inequality J_lambda(u0 + r u_{epsilon,eta}) < c_lambda is the mechanism for obtaining the second solution, so either the theorem must be restricted to weights that are positive near the concentration point or the concentration point must be moved into the positive set of f with all estimates redone.","section":"Section 5, Proposition 5.2, inequality (5.5)"},{"comment":"After obtaining a critical point u0 of J_lambda, the paper introduces the positive-part functional J_lambda^+ and states that 'critical points of J_lambda are also critical points of J_lambda^+'. This is false in general: if phi is supported on {u0 < 0}, the derivative of J_lambda^+ at u0 contains no contribution from (u0^+)^{p-1} or (u0^+)^{2*-1}, while the derivative of J_lambda does. The subsequent test with phi = u0^- applies to critical points of J_lambda^+, not to the minimizer of J_lambda obtained above. A separate minimization of J_lambda^+ on its Nehari set, or an equivalent replacement argument, is needed to conclude that the solution is nonnegative. This affects the advertised conclusion 'nonnegative solutions' in both theorems.","section":"Section 4, proof of Theorem 1.1, nonnegativity argument"}],"minor_comments":[{"comment":"The formula for the unique critical point of m_u(t) is valid only for theta > 1; when theta = 1, m_u(t) has no positive critical point and the case should be treated separately.","section":"Section 3, before Proposition 3.1"},{"comment":"The proposition states that {u_k} is a Palais-Smale sequence, but the proof begins by assuming that {u_k} is bounded; in the applications boundedness follows from coercivity on N_lambda, so the statement should be amended accordingly.","section":"Section 3, Proposition 3.7"},{"comment":"The ball is written as B(0, xi) although xi is the function being constructed; this is presumably B(0, epsilon).","section":"Section 3, proof of Lemma 3.5"},{"comment":"The line 'Let 0 neq u in N_lambda subset N_lambda^+' is not correct, since N_lambda is not contained in N_lambda^+; the argument should start with u in N_lambda^+.","section":"Section 4, Lemma 4.1"},{"comment":"Inequality (5.3) is stated for p > 2, whereas p in (1,2) throughout the paper; if the inequality is intended for the critical exponent 2*, it should be relabeled so that it is not asserted for the concave exponent.","section":"Section 5, inequality (5.3)"},{"comment":"Lemma 5.1 is asserted without proof; since it supplies the endpoint needed to show that the path gamma(t) meets N_lambda^-, a proof or a precise adaptation of [41] to the mixed operator should be included.","section":"Section 5, Lemma 5.1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends the standard concave-convex critical Nehari-manifold machinery to a mixed local/nonlocal Kirchhoff problem with sign-changing weight under N+4s<6. That combination is new, and the functional setup follows the literature closely. If repaired, the result would be a solid contribution.\n\nTwo load-bearing gaps prevent me from accepting the theorems as written. First, in Lemma 3.5 the auxiliary function F_u(t,v) = t^2 rho(u-v)^2 - lambda t^p ∫ f|u-v|^p - t^{2*}∫|u-v|^{2*} omits the a-term and b-term of the Nehari condition, which should include a t^2 rho^2 + b t^{2theta} rho^{2theta}. So the implicit function theorem produces a map whose image lies in the Nehari set of the wrong functional, and Lemma 3.6's Palais-Smale sequence does not follow. Both existence proofs depend on that. Second, Proposition 5.2 uses f>0 on the support of the test function to drop the concave term, but the hypotheses only give f in L^{2*/(2*-p)} and sign-changing, with no positivity near the concentration point. The energy estimate is therefore not proved from assumptions.\n\nThere are smaller issues: Lemma 4.1 writes N_lambda subset N_lambda^+, and the proof of Lemma 3.6 is omitted. The citation pattern is fine.\n\nThe paper deserves a serious referee because the topic is timely and the flaws look repairable, but I would not accept it in current form. My recommendation: reject with invitation to revise, with the repairs made explicit.","headline":"New combination but two load-bearing gaps: the Nehari auxiliary map omits the Kirchhoff terms, and the second-solution energy estimate assumes f>0 near the concentration point.","tokens_in":848,"tokens_out":1144,"would_cite":false,"duration_ms":36318,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A01","35A15","35B33","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the mixed operator −Δ + (−Δ)^s with a sign-changing concave term, two distinct nonnegative solutions exist in low dimensions when N+4s<6 and the parameters are small.","keywords":["Mixed local and nonlocal operators","Kirchhoff type problem","Critical nonlinearity","Nehari manifold","sign-changing weight","multiplicity of solutions","concave-convex nonlinearity","fractional Laplacian"],"falsifier":"A direct sign check settles the scope of the proof: take $N=3$, $s=1/4$ (so $N+4s=4<6$), $p=3/2$, and $f \\equiv -1$ in the unit ball with $f \\equiv +1$ outside, which is an admissible sign-changing weight. On the bubble's support the quantity $(u_0 + r u_\\varepsilon)^p - u_0^p - p u_0^{p-1} r u_\\varepsilon$ is nonnegative for $p \\in (1,2)$, so $-\\tfrac{\\lambda}{p}\\int f(\\cdots)$ flips sign when $f<0$ there; computing the resulting term of order $\\lambda \\varepsilon^{N-(N-2)p/2}$ and comparing it with the leading negative term $-C\\varepsilon^{(N-2)/2}$ in equation (5.6) shows whether the sub-threshold bound $J_\\lambda(u_0 + r u_{\\varepsilon,\\eta}) < c_\\lambda$ survives. If it fails, Theorem 1.2 as stated does not follow from the given estimates for weights that are negative at the origin.","tokens_in":15834,"feed_emoji":"🧮","tokens_out":33564,"duration_ms":268034,"temperature":0.7,"pith_summary":"This paper tackles a boundary-value problem built from the mixed operator $-\\Delta + (-\\Delta)^s$ — the usual Laplace operator plus a fractional one — with a coefficient $a + b\\rho(u)^{2\\theta-2}$ in front that depends on the norm of the solution, in the spirit of Kirchhoff's vibrating-string model. The nonlinearity is the sum of a term at critical Sobolev growth, $|u|^{2^*-2}u$, and a lower-order concave term $\\lambda f(x)|u|^{p-2}u$ with $1 < p < 2$, where the weight $f$ is allowed to change sign. The paper claims two existence results: one nontrivial nonnegative solution for small $\\lambda$ in every dimension $N \\ge 3$, and — under the restriction $N+4s<6$, which confines the result to dimensions 3, 4, and 5 with suitably small $s$ — a second, distinct nonnegative solution for small $\\lambda$ and small $b$. Why it matters: the mixed operator's best Sobolev constant is known not to be attained, so the usual compactness machinery fails at the critical level, and the paper shows the two-solution concave–convex picture still survives.","feed_headline":"Critical Kirchhoff problem yields two solutions in dimensions 3-5","feed_subtitle":"Under N+4s<6, the mixed local-nonlocal operator admits two nonnegative solutions once λ and b are small.","key_machinery":"The argument turns on three objects. First, the mixed Sobolev space $X^{1,2}(\\Omega)$ with norm $\\rho(u) = (\\int_{\\mathbb{R}^N}|\\nabla u|^2\\,dx + [u]_s^2)^{1/2}$, whose sharp embedding constant $S_{N,s}(\\Omega)$ coincides with the classical $S_N$ and is not attained — a fact the paper imports, and the reason every energy estimate must land strictly below a critical threshold to recover compactness. Second, the Nehari manifold $\\mathcal{N}_\\lambda$, with its fibering-map decomposition into components $\\mathcal{N}^+_\\lambda$, $\\mathcal{N}^-_\\lambda$, $\\mathcal{N}^0_\\lambda$ corresponding to local minima, local maxima, and inflection points of the map $t \\mapsto J_\\lambda(tu)$; the extremal value $\\lambda^*$, a generalized Rayleigh quotient (Definition 3.2), marks the range $\\lambda < \\lambda^*$ in which $\\mathcal{N}^0_\\lambda$ is empty and the two components stay separated. Third, the concentrating test functions $u_{\\varepsilon,\\eta} = \\eta u_\\varepsilon / \\|\\eta u_\\varepsilon\\|_{L^{2^*}(\\Omega)}$, built from the canonical bubbles of the critical Sobolev embedding and cut off near the origin; the expansion of $J_\\lambda(u_0 + r u_{\\varepsilon,\\eta})$ along the crossing path yields the sub-threshold bound that drives the proof of the second solution.","core_discovery":"The central discovery is that the mixed operator $-\\Delta + (-\\Delta)^s$ inherits the full two-solution structure of the concave–convex problem despite its degenerate variational geometry. The author shows that the Nehari manifold $\\mathcal{N}_\\lambda$ splits into two nonempty components $\\mathcal{N}^+_\\lambda$ and $\\mathcal{N}^-_\\lambda$ for every $\\lambda$ below an extremal value $\\lambda^*$; the inflection component $\\mathcal{N}^0_\\lambda$ is empty there, so minimizers on each component are genuine weak solutions. The first solution is the negative-energy minimizer on $\\mathcal{N}^+_\\lambda$; the second is the minimizer on $\\mathcal{N}^-_\\lambda$, whose energy lies below the compactness threshold $c_\\lambda$ of Proposition 3.7. To cross from the first solution's component into $\\mathcal{N}^-_\\lambda$ the author uses a continuous path $u_0 + t\\, l\\, u_{\\varepsilon,\\eta}$ built from the canonical concentrating bubbles of the critical Sobolev problem, and the energy expansion along this path (with $b$ taken of order $\\varepsilon^q$, $q > N-2$) shows the infimum on $\\mathcal{N}^-_\\lambda$ falls below $c_\\lambda$ exactly when $N+4s<6$. Since $\\mathcal{N}^+_\\lambda$ and $\\mathcal{N}^-_\\lambda$ are disjoint and closed, the two minimizers are distinct, and both are shown to be nonnegative by testing the weak formulation against negative parts.","pith_inferences":["Editorial: the invoked positivity of $f$ on the bubble's support is not part of Theorem 1.2's hypotheses; recentering the bubble at a point where the positive part $f^+$ has positive density would extend the argument to any sign-changing $f$, but the paper includes no such translation step.","Editorial: the condition $N+4s<6$ is exactly the requirement $\\min\\{N-2, 2-2s\\} > (N-2)/2$, i.e., that the positive error terms in the bubble expansion decay faster than the negative correction; in dimension 6 and above that comparison fails, so the dimension bound is structural rather than a technical afterthought.","Editorial: Remark 4 suggests replacing the critical exponent by $q \\in (2\\theta, 2^*)$; with compact embedding replacing the threshold analysis, the same two-branch machinery should give multiplicity for all $\\lambda \\in (0, \\Lambda^* + \\varepsilon)$ and in every dimension, which is a testable extension of the paper's own framework.","Editorial: the same decomposition should carry over to other monotone Kirchhoff coefficients $M(t)$ with comparable growth, since the argument only uses the monotonicity of the map $h(t)$ in Proposition 3.1 and the shape of the extremal value $\\lambda^*$."],"forward_implications":["In dimensions $N = 3, 4, 5$ with $N+4s<6$, the problem $(P_\\lambda)$ has two distinct nontrivial nonnegative weak solutions for every $\\lambda \\in (0, \\Lambda_{00})$ and all sufficiently small $b$.","The first solution exists in every dimension $N \\ge 3$ for $\\lambda \\in (0, \\Lambda_0)$; the restriction $N+4s<6$ enters only through the second solution's energy estimate.","The second solution sits on the local-maximum branch $\\mathcal{N}^-_\\lambda$ of the Nehari manifold while the first sits on the local-minimum branch $\\mathcal{N}^+_\\lambda$, so the two solutions differ in variational character and cannot coincide.","Both solutions are nonnegative, obtained by testing the weak form against negative parts of the solution, even though the fractional kinetic term does not commute with taking absolute values.","For $\\lambda \\ge \\lambda^*$ the Nehari set ceases to be a manifold ($\\mathcal{N}^0_\\lambda$ becomes nonempty), so the multiplicity statement is restricted to the sub-extremal parameter range."],"supporting_citations":[{"why":"Supplies the two facts the critical analysis leans on: the mixed Sobolev constant $S_{N,s}(\\Omega)$ equals the classical $S_N$ and is never attained, plus the bubble estimates used in Proposition 5.2.","marker":"[11]"},{"why":"Provides the sign-changing-weight two-solution scheme on which Lemma 5.1 and the crossing between the components $\\mathcal{N}^+_\\lambda$ and $\\mathcal{N}^-_\\lambda$ are modelled.","marker":"[41]"},{"why":"The fractional Kirchhoff prototype with critical indefinite nonlinearity; its Proposition 3.8 is the template for the Ekeland-type minimizing sequence in Lemma 3.6.","marker":"[21]"},{"why":"The weak-convergence decomposition identity used in Proposition 3.7 to split the norm of a Palais–Smale sequence into a limiting part and a vanishing remainder.","marker":"[13]"},{"why":"The concave–convex prototype that defines the nonlinearity class and the two-solution profile this paper extends to the mixed operator.","marker":"[2]"},{"why":"Gives the result that a local minimizer of $J_\\lambda$ on the Nehari manifold is a genuine critical point when $\\mathcal{N}^0_\\lambda$ is empty (Lemma 3.3).","marker":"[14]"},{"why":"Defines the extremal value of the Nehari manifold as an infimum of the generalized Rayleigh quotient, used to prove $\\mathcal{N}^0_\\lambda = \\emptyset$ for $\\lambda < \\lambda^*$.","marker":"[26]"}],"fun_headline_variants":["Mixed Kirchhoff: two solutions when N+4s<6","Two critical solutions for mixed Kirchhoff under N+4s<6","Two nonnegative solutions for mixed Kirchhoff under critical growth","When N+4s<6, mixed Kirchhoff has two solutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the energy can be pushed below the compactness threshold assumes the sign-changing weight $f$ is positive on the support of the concentrating test function, but the theorem's hypothesis only places $f$ in a Lebesgue space and allows it to change sign without any local-positivity condition near the concentration point.","fun_headline_variants_meta":{"raw":{"variants":["Mixed Kirchhoff: two solutions when N+4s<6","Two critical solutions for mixed Kirchhoff under N+4s<6","Two nonnegative solutions for mixed Kirchhoff under critical growth","When N+4s<6, mixed Kirchhoff has two solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001162,"raw_usage":{"total_tokens":4785,"prompt_tokens":894,"completion_tokens":3891,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":3816}},"tokens_in":510,"tokens_out":3891,"duration_ms":24307,"temperature":1.0,"reasoning_tokens":3816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:35:34.057475+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct sign check settles the scope of the proof: take $N=3$, $s=1/4$ (so $N+4s=4<6$), $p=3/2$, and $f \\equiv -1$ in the unit ball with $f \\equiv +1$ outside, which is an admissible sign-changing weight. On the bubble's support the quantity $(u_0 + r u_\\varepsilon)^p - u_0^p - p u_0^{p-1} r u_\\varepsilon$ is nonnegative for $p \\in (1,2)$, so $-\\tfrac{\\lambda}{p}\\int f(\\cdots)$ flips sign when $f<0$ there; computing the resulting term of order $\\lambda \\varepsilon^{N-(N-2)p/2}$ and comparing it with the leading negative term $-C\\varepsilon^{(N-2)/2}$ in equation (5.6) shows whether the sub-threshold bound $J_\\lambda(u_0 + r u_{\\varepsilon,\\eta}) < c_\\lambda$ survives. If it fails, Theorem 1.2 as stated does not follow from the given estimates for weights that are negative at the origin.","supporting_citations":[{"cited_title":"Wu, On semilinear elliptic equations involving c ritical Sobolev exponents and sign-changing weight function, Commun","cited_arxiv_id":null,"evidence_quote":"Provides the sign-changing-weight two-solution scheme on which Lemma 5.1 and the crossing between the components $\\mathcal{N}^+_\\lambda$ and $\\mathcal{N}^-_\\lambda$ are modelled."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The fractional Kirchhoff prototype with critical indefinite nonlinearity; its Proposition 3.8 is the template for the Ekeland-type minimizing sequence in Lemma 3.6."},{"cited_title":"Brezis, E","cited_arxiv_id":null,"evidence_quote":"The weak-convergence decomposition identity used in Proposition 3.7 to split the norm of a Palais–Smale sequence into a limiting part and a vanishing remainder."},{"cited_title":"Ambrosetti, H","cited_arxiv_id":null,"evidence_quote":"The concave–convex prototype that defines the nonlinearity class and the two-solution profile this paper extends to the mixed operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the result that a local minimizer of $J_\\lambda$ on the Nehari manifold is a genuine critical point when $\\mathcal{N}^0_\\lambda$ is empty (Lemma 3.3)."},{"cited_title":"Il’yasov, On extreme values of Nehari manifold metho d via nonlinear Rayleigh’s quotient, Topol","cited_arxiv_id":null,"evidence_quote":"Defines the extremal value of the Nehari manifold as an infimum of the generalized Rayleigh quotient, used to prove $\\mathcal{N}^0_\\lambda = \\emptyset$ for $\\lambda < \\lambda^*$."}],"review_version":1}